Two new regularization methods for solving sideways heat equation

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1 Nguyen and uu Journal of Inequalities and Applications 015) 015:65 DOI /s R E S E A R C H Open Access Two new regularization methods for solving sideways heat equation Huy Tuan Nguyen 1* and Vu Cam Hoan uu * Correspondence: nguyenhuytuan@tdt.edu.vn 1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article Abstract We consider a non-standard inverse heat conduction problem in a bounded domain which appears in some applied subjects. We want to know the surface temperature in a body from a measured temperature history at a fied location inside the body. This is an eponentially ill-posed problem in the sense that the solution if it eists) does not depend continuously on the data. In this paper, we introduce the two new classes of quasi-type methods and iteration methods to solve the problem and prove that our methods are stable under both aprioriand a posteriori parameter choice rules. An appropriate selection of a parameter in the scheme will get a satisfactory approimate solution. Furthermore, if we use the discrepancy principle we can avoid the selection of the aprioribound. MSC: 35K05; 35K99; 47J06; 47H10 Keywords: Cauchy problem; sideways heat equation; ill-posed problem; error estimates 1 Introduction In this paper, we want to determine the surface temperature u, t) for0< < from well-known temperature measurements u, t) = gt) when u, t) satisfies the following system: u t = u, 0< <,0 t π, u, t)=gt), 0 t π, u,0)=0, 0 t π. 1) In order to guarantee the uniqueness of the solution, here we assume that u be bounded as +. The functions g, h given in 0, π) are to be noised. Physically, we only obtain the measured Cauchy data with measurement errors. This is a severely ill-posed problem: any small perturbation in the observation data can cause large errors in the solution u, t) for [0,). Therefore, most classical numerical methods often fail to give an acceptable approimation of the solution. Thus it requires regularization techniques to stabilize the numerical computations [1]. The problem has a long history and has attracted to many authors. In several engineering contets, it is sometimes necessary to determine the surface temperature and heat flu in a body from a measured temperature history at a fied location inside the body [, 3]. The physical situation at the surface may be unsuitable for attaching a sensor, or the accuracy 015 Nguyen and uu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution icense which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

2 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page of 17 of a surface measurement may be seriously impaired by the presence of the sensor. The special case of estimating a surface condition from interior measurements has come to be known as the inverse heat conduction problem IHCP). In recent years, IHCP have been researched by many authors and some valuable method are proposed, such as the difference regularization method [4], the Fourier method [5], quasi-reversibility method [, 6],wavelet and wavelet-galerkin method [7],aspectralregularization method [7, 8], etc. Most of the papers involving regularization methods for sideways heat are devoted to giving a convergence analysis with the regularization parameter chosen by an apriori choice rule, under which some aprioriinformation on the unknown eact solution which is to be simulated is often used. However, in general, the aprioribound M cannot be known eactly in practice, and working with a wrong constant M may lead to the bad regularized solution. In the present paper, we proposed two new regularization method: a modified quasi-boundary value method and an iteration method for solving Problem 1). This method has been used by Deng and iu to deal with the sideways parabolic equation [9], but here we give another way to choose the regularization parameter by an aposteriori rule. Moreover, we shall give a criterion for making an aposteriorichoice of the regularization parameter. To the authors knowledge, there are very few papers for choosing the regularization parameter by the aposteriorirulefor this sideways heat. This paper is constructed as follows: in Section, we propose two new regularization method. Convergence estimates under priori and posteriori assumptions are given in Section 3 and Section 4. Finally, we draw a conclusion to our method. Formulation of the problem and a quasi-boundary value regularization method Suppose g is for measured data in 0, π) and satisfies g g, ) 0,π) where > 0 represents a noisy level. For f 0, π), we have the Fourier series f t)= f k e ikt, 3) where the Fourier coefficient is f k = 1 π f t)e ikt dt. 4) π 0 The norm of f in is f 0,π) =π f k. 5) The principal value of ik is + i) ik = 1, k 0, 6) 1 i), k <0.

3 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 3 of 17 Suppose that the solution of the problem is represented as a Fourier series u, t)= u k )e ikt, 7) with u k )= 1 π u, t)e ikt dt. 8) π 0 et F, t)= F k)e ikt with F k )= 1 π F, t)e ikt dt. 9) π 0 Then by taking the partial derivative with respect to t and,we obtain u t, t)= u k )ike ikt, u, t)= d d u k)e ikt. 10) The main equation u t = u leads to u k )ike ikt = d d u k)e ikt. Combining the Cauchy data g, h, we have the following systems of second ordinary equations: where d d u k) iku k )=0, u k )=g k, 11) g k = gt), e ikt = 1 π gt)e ikt dt. π 0 The solution of 11)is u k )=C k ep ik)+d k ep ik). 1) Since u, t) isboundedwhen +, weconcludethatc k = 0. It follows from u k )= D k ep ik)=g k that D k = g k ep ik). This leads to the formula of u k : u k )=ep ik ) ) g k. 13)

4 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 4 of 17 The eact form of u is u, t)= ep ik ) ) gt), e ikt e ikt, 0. 14) emma 1 For y R, we have e iky = e y. 15) Proof We can rewrite the term e iky by e iky = e 1+i) y = e y ) ) y y cos + sin = e y. 16) 3 Quasi-boundary value method for a sideways heat equation Define the Sobolev space H s 0, π)as u0, t) H s 0,π) = 1+k ) s u0, t), e ikt. In this section, we present a QBV regularization problem as follows: u β) t = u β), 0< <,0 t π, u β), t)=b g t), 0 t π, u β),0)=0, 0 t π, 17) with the additive information that u, t)boundedwhen +. And B is defined as B g t)= ep ) β)+ep ) g t), e ikt e ikt. 18) 3.1 A priori parameter choice rule Theorem 1 et u be an eact solution to Problem 1). et g be as in ). 1) If u0, t) 0,π) M then by β)=, we have M u β), ) u, ) 0,π) M1. ) If u0, t) H s 0,π) M 1 then by β)= k M 1, 0<k <1, we have where u β), ) u, ) ) 1 ) k 0,π) 1 k s + M 1 As), M 1 lnm 1 /) As)= s s) s e 1 s 1+ s).

5 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 5 of 17 Proof Proof of Part 1. The solution of this regularized problem is u β), t)= ep ) ep ik )) β)+ep ) g t), e ikt e ikt. 19) et v β) obey v β), t)= ep ) ep ik )) β)+ep ) gt), e ikt e ikt. Step 1. Estimate u β) v β) 0,π). We have u β) v β) 0,π) =π ep ) ) ep ik ) β)+ep ) g t) gt), e ikt =π ep ) [β)+ep )] g t) gt), e ikt β) g t) gt) β). 0) 0,π) This implies that u β) v β) 0,π) β) 1 = M 1. 1) Step. Estimate v β) u 0,π).Usingu0, t), e ikt = ep ik)gt), e ikt,wehave v β) u 0,π) =π β) ep ) 1+βep ) ep ik ) ) gt), e ikt From β)= M,weobtain v β) u 0,π) M. =π β ep ik) u0, t), e ikt [β)+ep )] = β) π [ ep ) ] u0, t), e ikt β)+ep ) β) β) u0, t) β) 0,π) M. )

6 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 6 of 17 This implies that v β) u 0,π) M1. 3) Combining )and3), we obtain u β) u 0,π) u β) v β) 0,π) + v β) u 0,π) M1. Proof of Part. emma et s >0,X 0. Then for 0<β <1,we have β s s e 1 s 1+ s) ) s. 1 + )s β + ep )) ln1/β) Proof Case 1. [0, 1 ]. It is clear to see that β 1 + )s β + ep )) β 1 + )s ep ) β ep ) eβ. From the inequality β s e )s 1 ln1/β) )s,weget β s s e 1 s 1+ s) ) s. 4) 1 + )s β + ep )) ln1/) Case. > 1 ep.sety = β β 1 + )s β + ep ) = )).Thenweobtain β β + βy lnβy ) ) s ) 1 s = 1+Y lnβy ) ) 1 s ) lnβ) s = 1+Y ln1/β) lnβy ) ) s ) 1 lnβ) s =. 5) ln1/β) 1+Y lnβy ) We continue to estimate the term 1 1+Y lnβ) lnβy ) )s. If 0 < Y 1then0< lnβ)< lnβy ), thus ) 1 lnβ) s <1, 6) 1+Y lnβy )

7 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 7 of 17 else if Y >1thenln Y >0andlnβY )= Therefore ln Y 1+lnβY )) 0. This implies that 0< ln β lnβy ) < ln β lnβy ) <1+ln Y. Hence, in this case, we get ) 1 lnβ) s 1 + ln Y )s < =1+ln Y ) s Y 1. 1+Y lnβy ) Y < 1 due to the assumption Set gy )=1+ln Y ) s Y 1 for Y > e 1. Taking the derivative of this function, we get 1, ). g Y )=1+ln Y ) s 1 Y s 1 ln Y ). The function g has a maimum at the point Y 0 such that g Y 0 ) = 0. This implies that Y 0 = e s 1. And therefore sup1 + ln Y ) s Y 1 gy 0 )=s s e 1 s. 7) Y 1 Since 6)and7)hold,wehave ) 1 lnβ) s s s e 1 s. 1+Y lnβy ) From 5), we get ) β T s s s e 1 s s s e 1 s 1+T s) ) T s. 8) 1 + )s β + ep )) ln1/β) ln1/β) We have v u 0,π) =π π [ β ep ) β + ep ) β 1 + )4s β + ep s) 4s e 4s 1+ s) ln1/β) ] u0, t), e ikt )) ) 4s π 1+ ) 4s u0, t), e ikt ) 4s 1+ u0, t), e ikt 4 s s) 4s e 4s 1+ s) ) 4s π 1+k ) s u0, t), e ikt ln1/β) ) 4s As) u0, t) ln1/β) H s 0,π), 9)

8 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 8 of 17 where As)= s s) s e 1 s 1+ s). This leads to v β) u s 0,π) ln1/β)) As) u0, t) H s 0,π) ) s M 1 As). 30) ln1/β) Combining 0)and30), we obtain u β) u 0,π) u β) v β) 0,π) + v β) u 0,π) ) β) s 1 + M 1 As). ln1/β) 3. A posteriori parameter choice rule Theorem Suppose that 0< < g 0,π). Choose m >1such that 0<m < g 0,π). Then there eists a unique number β) such that u β), t) g = m. 31) 0,π) Furthermore, if u0, t) 0,π) Mthenbyβ)=, we have the following estimate: M u β), ) u, ) [ ] m 1 0,π) m 1 M [ ] m +1). Proof The following results are straightforward. emma 3 Set ρβ)= u β), t) g 0,π). If 0< < g 0,π) then ρ is a continuous strictly increasing function and satisfies a) lim β 0 ρβ)=0, b) lim β + ρβ)= g 0,π). From this lemma, it follows that there eists a unique number β) satisfying31). Set z β), t)=u, t) u β), t). Then z β) satisfies the heat equation z β) t = z β). 3) From the formula of u and u β),weget z β), t)=u β), t) u, t)= ep ik ) ) R k β, g, g ) e ikt,

9 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 9 of 17 where R k β, g, g ) ep = ) β)+ep ) g t), e ikt gt), e ikt. Then using the Hölder inequality, we get z β), t) 0,π) = ep 1 )) Rk β, g, g ) Rk β, g, g ) This implies that [ ep 1 [ Rk β, g, g ) ) ] 1 )) Rk β, g, g ) ) 1 ] 1 1 [ = ep ) Rk β, g, g ) ] 1 [ Rk β, g, g ) ] = z β) 0, t) 0,π) z β), t) 0,π) z β) 0, t) 1 Proof of Part a). It is easy to see that 0,π) z β), t) z β), t) 0,π) = u β), t) u, t) 0,π) 0,π). z β), t). 33) 0,π) u β), t) g 0,π) + g g m +1) 34) 0,π) and z β) 0, t) 0,π) u0, t) 0,π) + u β) 0, t) 0,π). 35) We have m = u β), t) g t) 0,π) β) ep = ) g t), e ikt e 1+β) ep 0,π) ikt ) β) ep ) g t) gt), e ikt e 1+β) ep 0,π) ikt ) + β) gt), e ikt e ikt β)+ep 0,π) )

10 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 10 of 17 g t) gt), e ikt e ikt + gt), β) ep 0,π) ) e ikt e ikt 0,π) = g g 0,π) + β) u0, t) 0,π) + β) u0, t) 0,π). This implies that β) 1 u0, t) m 1 0,π). 36) It follows from 19)that u β) 0, t) 0,π) = Combining 36) and37), we obtain ep ) ep ik) g t), e ikt e β)+ep 0,π) ikt ) ep ) ep ik) g t) gt), e ikt e β)+ep 0,π) ikt ) + ep ) ep ik) gt), e ikt e β)+ep 0,π) ikt ) 1 g β) t) gt), e ikt e ikt 0,π) + epik) gt), e ikt e ikt 0,π) 1 g g β) 0,π) + u0, t) 0,π) β) + u0, t) 0,π). 37) u β) 0, t) ) 1 u0, 0,π) m 1 +1 t) 0,π) = m u0, t) m 1 0,π). 38) From 35)and38), we obtain z β) 0, t) 0,π) 1+ m ) u0, t) m 1 0,π) This together with 33)and34)leads to z β), t) 0,π) z β) 0, t) 1 0,π) z β), t) m 1 = u0, t) m 1 0,π). 39) 0,π) [ m 1 u0, t) 1 [ ] m 1 0,π)] m +1).

11 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 11 of 17 Hence u, t) u β), t) [ ] m 1)M 1 0,π) m +1). m 1 4 An iteration regularization method 4.1 A priori parameter choice To the authors knowledge, there are few papers for choosing the regularization parameter by aposteriorirule for the sideways heat equation. In this section, a new regularization method of iteration type for solving this problem will be given. We will use this method solving Problem 1), an aprioriand aposteriorirule for choosing regularization parameters and the Hölder-type error estimates are given. To solve the problem, we introduce the following iteration scheme: u n, t), e ikt =1 h) u n 1, t), e ikt + h ep ik ) ) g t), e ikt 40) with initial guess u 0, t), e ikt and 0 < h = e <1,whichplaysanimportantrolein the convergence proof. From the formula, it is easy to see that u n, t), e ikt =1 h) n u 0, t), e ikt n h) i h ep ik ) ) g t), e ikt i=0 =1 h) n u 0, t), e ikt + [ 1 1 h) n] ep ik ) ) g t), e ikt. 41) Therefore, the approimate solution of Problem 1) hasthe followingform: u n, t)= u n, t), e ikt e ikt. 4) We have the main theorem as follows. Theorem 3 et u, t) be the eact solution of Problem 1), and u n be its regularization approimation defined by 4) with u 0, t), e ikt =0.Assume that u0, t) 0,π) M for M >0and take n =[ M ], where [k] denotes the largest integer not eceeding k, then we have the estimate u n, ) u, ) 0,π) M1. emma 4 For 0 h 1 and n 1, the following inequalities hold: 1 h) n h 1 n +1, 1 1 h) n n. h emma 5 For 0 h 1, 0 α 1, and n 1, the following inequalities hold: 1 h) n h α 1 n +1) α,

12 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 1 of h) n n α. h α From u 0, t), e ikt =0,weseethat u n, t)= [ 1 1 h) n ] ep ik ) ) g t), e ikt e ikt. et v, t)begivenby v n, t)= [ 1 1 h) n ] ep ik ) ) gt), e ikt e ikt. We divide the argument into two steps. Step 1. Estimate u n, ) v n, ) 0,π).Wehave u n, ) v n, ) 0,π) =π [ 1 1 h) n ] ep ik ) ) g gt), e ikt [ π sup 1 1 h) n ] ) ep ik ) g gt), e ikt [ π sup 1 1 h) n ] ) ep ) g g 0,π). It follows from ep )) = h and g g 0,π) that u n, ) v n, ) 0,π) Because of emma 5,wehave u n, ) v n, ) n 0,π). [ π sup 1 1 h) n ] h. 0<h<1 Therefore u n, ) v n, ) 0,π) n1. 43) Step. Estimate v n, ) u, ) 0,π).Wehave v n, ) u, ) 0,π) =π 1 h) n ep ) ) gt), e ikt u0, t) 0,π) sup 1 h) n h 0<h<1 M n +1). This implies that v n, ) u, ) 0,π) Mn +1). 44)

13 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 13 of 17 Combining 43) and44), we obtain u n, ) u, ) 0,π) u, ) v, ) 0,π) + v, ) u, ) 0,π) n 1 + Mn +1). Due to n =[ M ], we have n M and n +1 M, and therefore u n, ) u, ) 0,π) M M ) ) M 1 + =M The discrepancy principle In this section, we discuss the aposterioristopping rule for iteration scheme 40) based on discrepancy principle of Morozov in the following form: g t) u β, t) = m, 45) 0,π) where m >1 is a constant and β denotes the regularization parameter. If we take u 0, t), e ikt =0,then45) can be simplified to the form g t) u β, t) 0,π) = g t), e ikt e ikt We have the main theorem as follows. 1 1 h) [ β] g t), e ikt e ikt 0,π) = h) 1 β g t), e ikt e ikt = m. 46) 0,π) Theorem 4 et u, t) be the eact solution of Problem 1), and u n be its regularization approimation defined by 4) with u 0, t), e ikt =0.If the a priori bound u0, t) 0,π) M is valid and the iteration 40) is stopped by the discrepancy principle 45), then u β, ) u, ) 0,π) EM1, where E =m +1) ) m 1. m 1 The following results are obvious. emma 6 Set Pβ)= g t) u β, t) 0,π) = 1 h) β g t) 0,π). emma 7 Setting w β, t)= [ ep ik ) ) gt), e ikt [ 1 1 h) n] ep ik ) ) g t), e ikt ] e ikt,

14 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 14 of 17 the following inequality holds: w β, t) 0,π) w β 0, t) 1 Proof We have 0,π) w β, t) 0,π). w β 0, t)= [ ep ik) gt), e ikt [ 1 1 h) n] ep ik) g t), e ikt ] e ikt and w β, t)= [ gt), e ikt [ 1 1 h) n] g t), e ikt ] e ikt. We have w β, t) 0,π) = ep ) )[ gt), e ikt [ 1 1 h) n] g t), e ikt ]. For brevity in this case, we set Q k h, n)=gt), e ikt [1 1 h) n ]g t), e ikt.thenusing the Hölder inequality, we get w β, t) 0,π) = ep 1 )) Q k h, n) Q k h, n) [ ep 1 )) Q k h, n) [ Q k h, n) ) ] 1 ) 1 ] 1 1 [ = ep ) Q k h, n) ] 1 [ Q k h, n) ] = w β 0, t) 0,π) w β, t) 0,π). This completes the proof of the lemma. emma 8 The following inequality holds: β M m 1. Proof It follows from 46)that m = h) 1 β g t), e ikt e ikt h) 0,π) 1 β g t) gt), e ikt e ikt 0,π) + h) 1 β gt), e ikt e ikt 0,π)

15 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 15 of 17 sup 1 h) β g 0<h<1 t) gt), e ikt e ikt 0,π) + sup 1 h) β gt), e ikt e ikt 0<h<1 0,π). Moreover, it is easy to see that 1 h) β 1 β and gt), e ikt e ikt 0,π) = ep ik) u0, t), e ikt e ikt 0,π) u0, t) 0,π) M. This leads to β + M m. This completes the proof of the lemma. emma 9 The following inequality holds: w β, ) m +1). 0,π) Proof Due to the triangle inequality, we have w β, ) 0,π) = gt), e [ ikt [ 1 1 h) n] g t), e ikt ] e ikt 0,π) gt) g t), e ikt e ikt 0,π) + h) 1 β g t), e ikt e ikt 0,π) g g 0,π) + + m =m +1). 1 h) β g t), e ikt e ikt 0,π) emma 10 The following inequality holds: w β 0, ) 0,π) mm m 1. Proof Due to the triangle inequality and h = e,wehave w β 0, ) 0,π) = ep ik) gt), e [ ikt [ 1 1 h) β] epik) g t), e ikt ] e ikt 0,π)

16 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 16 of 17 ep ik) gt), e [ ikt [ 1 1 h) n] epik) gt), e ikt ] e ikt 0,π) h) [[ β] epik) gt) g t), e ikt ] e ikt 0,π) 1 h) [ β epik) gt), e ikt ] e ikt 0,π) h) β gt) g t), e ikt e ikt. 47) h 0,π) Moreover, we have 1 h) [ β epik) gt), e ikt ] e ikt 0,π) ep ik) gt), e [ ikt ] e ikt 0,π) = u0, t) M, 48) 0,π) and using 1 1 h)β β,wehavetheestimate h 1 1 h) β gt) g t), e ikt e ikt h 0,π) Combining 47), 48), and 49), we obtain w β 0, ) 0,π) M + β. β gt) g t), e ikt e ikt 0,π) β g g β. 49) 0,π) emma 8 leads to w β 0, ) 0,π) M + M m 1 = Mm m 1. Now, we return to the proof of the theorem. Combining emma 7, emma 8, and emma 9,weobtain u, ) u, ) 0,π) = w β, ) 0,π) w β 0, t) 1 0,π) w β, t) 0,π) ) Mm 1 ) m +1) m 1 = EM 1, where E =m +1) ) m 1. m 1

17 Nguyenand uu Journal of Inequalities and Applications 015) 015:65 Page 17 of 17 Competing interests The authors declare that they have no competing interests. Authors contributions HTN and VCH organized and wrote this manuscript. VCH contributed to all the steps of the proofs in this research together. HTN participated in the discussion and corrected the main results. All authors read and approved the final manuscript. Author details 1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. Faculty of Basic Science, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam. Acknowledgements The authors wish to epress their sincere thanks to the anonymous referees and the handling editor for many constructive comments, leading to the improved version of this paper. The second author is supported by Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam. Received: September 014 Accepted: 14 January 015 References 1. Kirsch, A: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol. 10. Springer, New York 1996). Eldén, : Approimations for a Cauchy problem for the heat equation. Inverse Problems 3), ) 3. Qian, Z, Fu, C-, Xiong, X-T: A modified method for determining the surface heat flu of IHCP. Inverse Probl. Sci. Eng. 153), ) 4. Xiong, X-T, Fu, C-, i, H-F: Central difference method of a non-standard inverse heat conduction problem for determining surface heat flu from interior observations. Appl. Math. Comput. 173), ) 5. Xiong, X-T, Fu, C-, i, H-F: Fourier regularization method of a sideways heat equation for determining surface heat flu. J. Math. Anal. Appl. 3171), ) 6. iu, JC, Wei, T: A quasi-reversibility regularization method for an inverse heat conduction problem without initial data. Appl. Math. Comput. 193), ) 7. Reginska, T, Elden, : Solving the sideways heat equation by a wavelet-galerkin method. Inverse Probl. 134), ) 8. Elden,, Berntsson, F, Reginska, T: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput. 16), ) 9. Deng, YJ, iu, ZH: Iteration methods on sideways parabolic equations. Inverse Probl. 5, )

A truncation regularization method for a time fractional diffusion equation with an in-homogeneous source

$A truncation regularization method for a time fractional diffusion equation with an in-homogeneous source$ ITM Web of Conferences, 7 18) ICM 18 https://doi.org/1.151/itmconf/187 A truncation regularization method for a time fractional diffusion equation with an in-homogeneous source Luu Vu Cam Hoan 1,,, Ho